You don't move the foundation, you move the building.

How do you get in between the foundations and the building to lift it?

I'm not sure about the construction of this building in particular -- but generally speaking you support the building from an alternative location under it, then you can lift it off the foundation.

https://i.pinimg.com/736x/93/cd/bb/93cdbbe7117f16e89c74e4909...

break through at several strategic points where the building meets the foundation and place hydraulic jacks in the voids. Once enough jacks are in place and lifted, _they_ now serve as the support; the remaining parts of the building can be removed/disconnected from the original foundation.

Thanks for the explanation. This is on a much smaller scale but shows what you’ve described:

https://youtu.be/Qnr5v0farM0

To give a more concrete answer: you dig under a small section of footing. Small enough that the footing itself bridges the load across the gap. Then you install your first support element between the remaining undisturbed soil and the footing. Now that section is supported just as it was before, but by a jack, strut, wheel truck, or whatever instead of soil. Rinse and repeat until all the soil is gone.

For larger internal spans you usually install bracing elements like trusses or I-beams. This is done in a similar way: you hollow out a narrow strip, install the truss/beam, then that gives you enough leeway to continue digging on either side of it until you reach the location for the next truss/beam. Shims are used to ensure solid contact all along the length. The trusses/beams are anchored to the support elements.

At any one time only a very small part of the building's foundation is "floating" and unsupported.

Actually moving the building varies based on its size, weight, and other factors. You'd usually jack up the whole building to make room for the trucking elements which might be independently steerable wheels, one or more semi-trailers, or whatever. Then you lower the building onto the trucking element and it becomes a really big vehicle or trailer that you slowly and carefully move.

At the destination you usually pour new footings and/or piers in the same pattern as the original support elements so your truck bits can just drive right into place. Then you again just jacks to lift the building off the truck elements, move them out, then lower the building onto the new footings.

This is very simplified. A lot of engineering goes into making sure the building moves as a single piece without too much flexing (vertically or horizontally) and that the "road" it moves on can support the weight.